In order to reply, we must resort to the ancient Aristotelian distinction between the temporal order in which the logical dependence of propositions is discovered and the logical order of implications between propositions. There is no doubt that many of the axioms of mathematics are an expression of what we believe to be the truth concerning selected parts of nature, and that many advances in mathematics have been made because of the suggestions of the natural sciences. But there is also no doubt that mathematics as an inquiry did not historically begin with a number of axioms from which subsequently the theorems were derived. We know that many of the propositions of Euclid were known hundreds of years before he lived; they were doubtless believed to be materially true. Euclid’s chief contribution did not consist in discovering additional theorems, but in exhibiting them as part of a system of connected truths. The kind of question Euclid must have asked himself was: Given the theorems about the angle sum of a triangle, about similar triangles, the Pythagorean theorem, and the rest, what are the minimum number of assumptions or axioms from which these can be inferred? As a result of his work, instead of having what were believed to be independent propositions, geometry became the first known example of a deductive system. The axioms were thus in fact discovered later than the theorems, although the former are logically prior to the latter.

It is a common prejudice to assume that the logically prior propositions are “better known” or “more certain” than the theorems, and that in general the logical priority of some propositions to others is connected in some way with their being true. Axioms are simply assumptions or hypotheses, used for the purpose of systematizing and sometimes discovering the theorems they imply. It follows that axioms need not be known to be true before the theorems are known, and in general the axioms of a science are much less evident psychologically than the theorems. In most sciences, as we shall see, the material truth of the theorems is not established by means of first showing the material truth of the axioms. On the contrary, the material truth of axioms is made probable by establishing empirically the truth or the probability of the theorems.